332 Mr. J. W. L. Glaisher on [Nov, 20, 



For example, from the product 



T(a)r(a + l\ . . . r(a+— • (2r)«»-l)fiH-, 

 V nj X n J 



since, when n is great, 

 we derive the evaluation, 



Ca+l 



\ogT(x)dx=±log (27r) + a(loga — 1). 



The number of products that yield integrals of interest is not great, 

 and those just noticed are all that I remember to have seen applied to 

 this purpose. The transformation formulae in elliptic functions lead, 

 however, in this manner, immediately to definite integrals, as will 

 appear in the next section. 



§ 3. Writing sn, on, dn in place of sin am, cos am, A am, we have 

 (" Fundamenta Nova," p. 4G) : — 



{sn2 W sn4 W ...sn(^-l)^^^^(^y . . (2). 

 {cn2wcn4w . . . cn (n— l)w} 2 =Q^_y (3). 



{dn2wdn4o> . . . dn (n-l)u}*=(*^ (4). 



In the case of the first real transformation (n being an uneven 

 prime) 



^M' M" 



Thus, when n is infinite, 



» i T\r 2K . / i ttK! 



A=±7r, M= , A =i%— , 



4 - hi^L 



but A'=log-, so that X=e ' k , 



A, 



XV 



and we therefore have 



Io g p) = ^+>loK(|)-ilog(H, 



